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\begin{document}

\begin{center}
  \Large{1. Gyak.}
  
  {vektor-vektor fv-ek, diff. operátorok}
  \end{center}

\medskip

\begin{enumerate}
 \item $\underline{v}(\underline{x}):=\underline{v}(x, y, z)=x^2 y z \underline{i}-\frac{2x\sqrt{z}}{y^2} \underline{j}, \underline{x}_0=(-1, 1, 4)^\top, \tilde{\underline{x}} := (-0.99 ;1.02; 3.9)^\top $
  \begin{enumerate}
    \item $\underline{\underline{J}}_{\underline{v}}(\underline{x})=?$
	\item Adjon közelítést $\underline{\tilde{x}}$-ra a teljes differenciál segítségével!
  \end{enumerate}
 
 \item $f(x, y, z):=2x^2 y-\frac{y^2 x}{z^2}, \underline{x}_0:=(1, -1, 1)^\top$
   \begin{enumerate}
     \item $\operatorname{grad}f=?, (\operatorname{grad}f)(\underline{x}_0)=?$
	 \item $\triangle f=?, (\triangle f)(\underline{x}_0)=?$
   \end{enumerate}
 
 \item $\underline{v}(\underline{x})=(3x^3 y; -y^2 z; z^2 y x)^\top, \underline{x}_0=(1, 1, 1)$
   \begin{enumerate}
     \item $\operatorname{div} \underline{v}=?, (\operatorname{div} \underline{v})(\underline{x}_0)=?$
	 \item $\operatorname{rot} \underline{v}=?, (\operatorname{rot} \underline{v})(\underline{x}_0)=?$
	 \item $\underline{\triangle} \underline{v}=?, (\underline{\triangle} \underline{v})(\underline{x}_0)=?$
   \end{enumerate}
   
  \item $f(\underline{x}):= 3y^2 x+\frac{z^3}{\sqrt{x}}, \underline{v}(\underline{x}):=\underline{x} \cdot |\underline{x}|^2$
    \begin{enumerate}
	  \item $\operatorname{div} \operatorname{grad} f=?$
	  \item $\operatorname{rot} \operatorname{grad} f=?$
	  \item $\operatorname{div} \operatorname{rot} \underline{v}=?$
	  \item $\operatorname{rot} \operatorname{rot} \underline{v}=?$
	\end{enumerate}
	
  \item $\underline{a}=(1, 2, 3)^\top, \underline{v}(\underline{x}):=2 \underline{a} \times \underline{x}$
    %\begin{multicols}{3}
	\begin{enumerate}
	  \item $\underline{\underline{J}}_{\underline{v}}(\underline{x})=?$
	  \item $\operatorname{div} \underline{v}=?$
	  \item $\operatorname{rot} \underline{v}=?$
	\end{enumerate}
	%\end{multicols}
\end{enumerate}

\end{document}
